Final answer:
The question pertains to performing an integral by making a substitution to express the integrand as a rational function. The process involves finding a suitable substitution and following steps to simplify and evaluate the integral, which may sometimes include considerations of a curve or path geometry.
Step-by-step explanation:
The question involves evaluating an integral by making a substitution to express the integrand as a rational function. This technique is often used to simplify the integration process when dealing with functions that are not easily integrable in their original form. To do this, we would typically let one part of the integrand be equal to a new variable, which allows us to rewrite the rest of the integrand in terms of this new variable.
Without the specific integral provided, a detailed step-by-step method cannot be given. However, the steps would involve:
- Choosing an appropriate substitution that simplifies the integrand into a rational function.
- Differentiating the chosen expression to obtain the differential substitution.
- Substituting both the function and differential into the original integral.
- Integrating the resulting rational function.
- Finding the antiderivative and substituting back the original variable if necessary.
- Adding the constant of integration c.
If the integral was based on a specific curve or path, such as around an arc of constant radius r, further steps would be required to evaluate the line integral by considering the geometry and parametrization of the path.